--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/+sbp/+implementations/d4_compatible_halfvariable_2.m	Thu Sep 08 15:35:45 2016 +0200
@@ -0,0 +1,162 @@
+% Returns D2 as a function handle
+function [H, HI, D1, D2, D3, D4, e_1, e_m, M4, Q, S2_1,...
+    S2_m, S3_1, S3_m, S_1, S_m] = d4_compatible_halfvariable_2(m,h)
+    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+    %%% 4:de ordn. SBP Finita differens         %%%
+    %%% operatorer framtagna av Ken Mattsson    %%%
+    %%%                                         %%%
+    %%% 6 randpunkter, diagonal norm            %%%
+    %%%                                         %%%
+    %%% Datum: 2013-11-11                       %%%
+    %%%                                         %%%
+    %%%                                         %%%
+    %%% H           (Normen)                    %%%
+    %%% D1          (approx f?rsta derivatan)   %%%
+    %%% D2          (approx andra derivatan)    %%%
+    %%% D3          (approx tredje derivatan)   %%%
+    %%% D2          (approx fj?rde derivatan)   %%%
+    %%%                                         %%%
+    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+    % M?ste ange antal punkter (m) och stegl?ngd (h)
+    % Notera att dessa opetratorer ?r framtagna f?r att anv?ndas n?r
+    % vi har 3de och 4de derivator i v?r PDE
+    % I annat fall anv?nd de "traditionella" som har noggrannare
+    % randsplutningar f?r D1 och D2
+
+    % Vi b?rjar med normen. Notera att alla SBP operatorer delar samma norm,
+    % vilket ?r n?dv?ndigt f?r stabilitet
+
+    H=diag(ones(m,1),0);H(1,1)=1/2;H(m,m)=1/2;
+
+
+    H=H*h;
+    HI=inv(H);
+
+
+    % First derivative SBP operator, 1st order accurate at first 6 boundary points
+
+    q1=1/2;
+    Q=q1*(diag(ones(m-1,1),1)-diag(ones(m-1,1),-1));
+
+    %Q=(-1/12*diag(ones(m-2,1),2)+8/12*diag(ones(m-1,1),1)-8/12*diag(ones(m-1,1),-1)+1/12*diag(ones(m-2,1),-2));
+
+
+    e_1=zeros(m,1);e_1(1)=1;
+    e_m=zeros(m,1);e_m(m)=1;
+
+
+    D1=HI*(Q-1/2*e_1*e_1'+1/2*e_m*e_m') ;
+
+    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+
+
+    % Second derivative, 1st order accurate at first boundary points
+
+    % below for constant coefficients
+    % m1=-1;m0=2;
+    % M=m1*(diag(ones(m-1,1),1)+diag(ones(m-1,1),-1))+m0*diag(ones(m,1),0);M(1,1)=1;M(m,m)=1;
+    % M=M/h;
+    %D2=HI*(-M-e_1*S_1+e_m*S_m);
+
+    % Below for variable coefficients
+    % Require a vector c with the koeffients
+
+    S_U=[-3/2 2 -1/2]/h;
+    S_1=zeros(1,m);
+    S_1(1:3)=S_U;
+    S_m=zeros(1,m);
+    S_m(m-2:m)=fliplr(-S_U);
+
+    S_1 = S_1';
+    S_m = S_m';
+
+    M=sparse(m,m);
+    e_1 = sparse(e_1);
+    e_m = sparse(e_m);
+    S_1 = sparse(S_1);
+    S_m = sparse(S_m);
+
+    scheme_width = 3;
+    scheme_radius = (scheme_width-1)/2;
+    r = (1+scheme_radius):(m-scheme_radius);
+
+    function D2 = D2_fun(c)
+
+        Mm1 = -c(r-1)/2 - c(r)/2;
+        M0  =  c(r-1)/2 + c(r)   + c(r+1)/2;
+        Mp1 =            -c(r)/2 - c(r+1)/2;
+
+        M(r,:) = spdiags([Mm1 M0 Mp1],0:2*scheme_radius,length(r),m);
+
+
+        M(1:2,1:2)=[c(1)/2 + c(2)/2 -c(1)/2 - c(2)/2; -c(1)/2 - c(2)/2 c(1)/2 + c(2) + c(3)/2;];
+        M(m-1:m,m-1:m)=[c(m-2)/2 + c(m-1) + c(m)/2 -c(m-1)/2 - c(m)/2; -c(m-1)/2 - c(m)/2 c(m-1)/2 + c(m)/2;];
+        M=M/h;
+
+        D2=HI*(-M-c(1)*e_1*S_1'+c(m)*e_m*S_m');
+    end
+    D2 = @D2_fun;
+
+
+
+
+
+    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+
+
+    % Third derivative, 1st order accurate at first 6 boundary points
+
+    q2=1/2;q1=-1;
+    Q3=q2*(diag(ones(m-2,1),2)-diag(ones(m-2,1),-2))+q1*(diag(ones(m-1,1),1)-diag(ones(m-1,1),-1));
+
+    %QQ3=(-1/8*diag(ones(m-3,1),3) + 1*diag(ones(m-2,1),2) - 13/8*diag(ones(m-1,1),1) +13/8*diag(ones(m-1,1),-1) -1*diag(ones(m-2,1),-2) + 1/8*diag(ones(m-3,1),-3));
+
+
+    Q3_U = [0 -0.13e2 / 0.16e2 0.7e1 / 0.8e1 -0.1e1 / 0.16e2; 0.13e2 / 0.16e2 0 -0.23e2 / 0.16e2 0.5e1 / 0.8e1; -0.7e1 / 0.8e1 0.23e2 / 0.16e2 0 -0.17e2 / 0.16e2; 0.1e1 / 0.16e2 -0.5e1 / 0.8e1 0.17e2 / 0.16e2 0;];
+    Q3(1:4,1:4)=Q3_U;
+    Q3(m-3:m,m-3:m)=flipud( fliplr( -Q3_U ) );
+    Q3=Q3/h^2;
+
+
+
+    S2_U=[1 -2 1;]/h^2;
+    S2_1=zeros(1,m);
+    S2_1(1:3)=S2_U;
+    S2_m=zeros(1,m);
+    S2_m(m-2:m)=fliplr(S2_U);
+    S2_1 = S2_1';
+    S2_m = S2_m';
+
+
+
+    D3=HI*(Q3 - e_1*S2_1' + e_m*S2_m' +1/2*S_1*S_1' -1/2*S_m*S_m' ) ;
+
+    % Fourth derivative, 0th order accurate at first 6 boundary points (still
+    % yield 4th order convergence if stable: for example u_tt=-u_xxxx
+
+    m2=1;m1=-4;m0=6;
+    M4=m2*(diag(ones(m-2,1),2)+diag(ones(m-2,1),-2))+m1*(diag(ones(m-1,1),1)+diag(ones(m-1,1),-1))+m0*diag(ones(m,1),0);
+
+    %M4=(-1/6*(diag(ones(m-3,1),3)+diag(ones(m-3,1),-3) ) + 2*(diag(ones(m-2,1),2)+diag(ones(m-2,1),-2)) -13/2*(diag(ones(m-1,1),1)+diag(ones(m-1,1),-1)) + 28/3*diag(ones(m,1),0));
+
+    M4_U=[0.13e2 / 0.10e2 -0.12e2 / 0.5e1 0.9e1 / 0.10e2 0.1e1 / 0.5e1; -0.12e2 / 0.5e1 0.26e2 / 0.5e1 -0.16e2 / 0.5e1 0.2e1 / 0.5e1; 0.9e1 / 0.10e2 -0.16e2 / 0.5e1 0.47e2 / 0.10e2 -0.17e2 / 0.5e1; 0.1e1 / 0.5e1 0.2e1 / 0.5e1 -0.17e2 / 0.5e1 0.29e2 / 0.5e1;];
+
+
+    M4(1:4,1:4)=M4_U;
+
+    M4(m-3:m,m-3:m)=flipud( fliplr( M4_U ) );
+    M4=M4/h^3;
+
+    S3_U=[-1 3 -3 1;]/h^3;
+    S3_1=zeros(1,m);
+    S3_1(1:4)=S3_U;
+    S3_m=zeros(1,m);
+    S3_m(m-3:m)=fliplr(-S3_U);
+    S3_1 = S3_1';
+    S3_m = S3_m';
+
+    D4=HI*(M4-e_1*S3_1'+e_m*S3_m'  + S_1*S2_1'-S_m*S2_m');
+end